List of top Linear Algebra Questions on Group Theory asked in IIT JAM MA

Let \( F = \{\omega \in \mathbb{C} : \omega^{2020} = 1\}. \) 

Consider the groups \[ G = \left\{ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix} : \omega \in F, z \in \mathbb{C} \right\} \text{and} H = \left\{ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} : z \in \mathbb{C} \right\} \] under matrix multiplication. 

Then the number of cosets of \( H \) in \( G \) is 

Consider the following group under matrix multiplication: 

 \[ H = \left\{ \begin{bmatrix} 1 & p & q \\ 0 & 1 & r \\ 0 & 0 & 1 \end{bmatrix} : p, q, r \in \mathbb{R} \right\}. \] 

Then the center of the group is isomorphic to 

Let \( S^1 = \{z \in \mathbb{C} : |z| = 1\} \) be the circle group under multiplication and \( i = \sqrt{-1}. \) Then the set \( \{\theta \in \mathbb{R} : (e^{i2\pi\theta}) \text{ is infinite}\} \) is